Prove termination of diff algorithm implementation - #37
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The manhattan distance from a node towards the algorithm's end point (lena, lenb) is used as termination metric for `addsnake`. Phantom parameters for both input lengths are introduced to provide them as arguments to the metric. The diagonal predicate `DiagPred` is a refinement type alias encoding the condition of an equality predicate (such as `canDiag`) to enter the recursive call inside `addsnake`. This allows Liquid Haskell to know that both coordinates are smaller than its corresponding input length, those fulfilling `manhattanDistance` preconditions. `dstep` is extended to provide the required phantom parameters and is temporarily ignored because proving node coordinates in a wave front are within bounds (`manhatanDistance` preconditions) requires discarting out-of-bounds nodes. This is implemented as an optimization in a following commit.
A `_wfDistanceToGoal` function is defined to be used as termination metric for `ses`. In this commit, `dstep` is specified and checked to reduce it from input to output. The wave front diagonal condition is changed to look at the head of the node list instead of the diagonal edit distance parameter, and its nodes are specified to be within bounds. Indeed, now that wave fronts are trimmed down to be within bounds, we can no longer guarantee that the first node's diagonal matches the edit distance. The `stepAndMerge` specification is strengthen to preserve this new variant of wave front diagonal invariant: With the current optimization of `dstep`, wave fronts don't necessarily grow, but the 2-step specing is preserved.
The implementation of `ses` changes from a `dropWhile` driven search for the algorithm's end point in a lazy stream composed of all wave front nodes, to the explicit recursion of a wave-front wise search for such end point. This change was designed to allow a termination proof using Liquid Haskell: by inspecting each wave front separately, instead of all concatenated together in an infinite stream, we can define a wave front metric as its minimum distance to the endpoint and show it is reduced by the recursive calls (to `dstep`). Performance-wise, we get a small optimization of the benchmark of ~12%
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…eason#28) The optimization introduced in `dstep` (narrowing the wave front to only within bound nodes) solves the problem this additional equations addressed in seereason@33bf8bc They are removed to avoid unnecessary complexity.
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Co-authored-by: Facundo Domínguez <facundominguez@gmail.com>
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The introduced benchmark makes the improvement of the `dstep` refactoring (dropping out-of-bounds nodes) more noticeable.
A definition less, plus upstream has an optimization for `const` in ucsd-progsys/liquidhaskell#2732 that could turn out to be useful (currently is makes no difference because lemmas are not being composed).
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| -- The next node being on the right border means all | ||
| -- previous nodes cannot compete to the endpoint, because their | ||
| -- children require more steps to cross the next node's diagonal. | ||
| -- HACK: However, we keep @prev@'s vertical child node to preserve | ||
| -- the '_wfDiags' invariant at a negligible performance penalty. |
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| -- The next node being on the right border means all | |
| -- previous nodes cannot compete to the endpoint, because their | |
| -- children require more steps to cross the next node's diagonal. | |
| -- HACK: However, we keep @prev@'s vertical child node to preserve | |
| -- the '_wfDiags' invariant at a negligible performance penalty. | |
| -- The next node being on the right border means that | |
| -- 'furthestReaching' would produce a node outside the grid, | |
| -- However, we keep @prev@'s vertical child node to preserve | |
| -- the '_wfDiags' invariant at a negligible performance penalty. | |
| -- See issue BLAH for alternatives. |
We need issue BLAH. Or perhaps a note in the code discussing the balance.
Co-authored-by: Facundo Domínguez <facundominguez@gmail.com>
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